A084860 Expansion of (1 - 2*x + 2*x^2 - x^3)/(1 - 2*x)^2.

1, 2, 6, 15, 36, 84, 192, 432, 960, 2112, 4608, 9984, 21504, 46080, 98304, 208896, 442368, 933888, 1966080, 4128768, 8650752, 18087936, 37748736, 78643200, 163577856, 339738624, 704643072, 1459617792, 3019898880, 6241124352, 12884901888, 26575110144, 54760833024

Offset

0,2

Comments

Partial sums are A084858.

Partial sums of A084860.

For n >= 2, a(n) is the total number of ones in runs of ones of length 3 over all binary strings of length n+2. - Félix Balado, Sep 29 2025

Links

Michael De Vlieger, Table of n, a(n) for n = 0..3311

Félix Balado and Guénolé C. M. Silvestre, Systematic Enumeration of Fundamental Quantities Involving Runs in Binary Strings, arXiv:2602.10005 [math.CO], 2026. See p. 111.

Index entries for linear recurrences with constant coefficients, signature (4,-4).

Formula

a(0) = 1, a(n+1) = 3*2^(n-2)*(n+3) - 0^n/4.

Equals binomial transform of nonzero terms of A026741: (1, 1, 3, 2, 5, 3, 7, 4, ...). - Gary W. Adamson, Apr 25 2008

Equals row sums of triangle A139633. - Gary W. Adamson, Apr 27 2008

E.g.f.: (3*(x+1)*exp(2*x) - x + 1)/4. - Amiram Eldar, Feb 19 2026

Mathematica

CoefficientList[Series[(1-2x+2x^2-x^3)/(1-2x)^2, {x, 0, 50}], x]  (* Harvey P. Dale, Mar 30 2011 *)

Crossrefs

Cf. A026741, A084858, A084860, A139633.

Sequence in context: A136302 A116404 A301600 * A084798 A215149 A017923

Adjacent sequences: A084857 A084858 A084859 * A084861 A084862 A084863

Keyword

easy,nonn

Author

Paul Barry, Jun 12 2003

Status

approved